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#[global ] Unset Universe Checking .
(*******************************************************************
Limits in bicategories of structured categories
We look at terminal objects, products, pullbacks, and
Eilenberg-Moore objects
Contents
1. Limits of categories with a terminal objects
2. Limits of categories with products
3. Limits of categories with pullbacks
4. Limits of categories with finite limits
5. Limits of categories with initial objects
6. Limits of categories with coproducts
*******************************************************************)
Require Import UniMath.Foundations.All.Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ = _" was already used in scope type_scope.
[notation-overridden,parsing,default]Notation "_ <-> _" was already used in scope
type_scope. [notation-overridden,parsing,default]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Import UniMath.MoreFoundations.All.Notation "_ ≠ _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ ≠ _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.Categories.StandardCategories.
Require Import UniMath.CategoryTheory.Categories.EilenbergMoore.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.CategoryTheory.IsoCommaCategory.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.Limits.Terminal.
Require Import UniMath.CategoryTheory.Limits.BinProducts.
Require Import UniMath.CategoryTheory.Limits.Pullbacks.
Require Import UniMath.CategoryTheory.Limits.Initial.
Require Import UniMath.CategoryTheory.Limits.BinCoproducts.
Require Import UniMath.CategoryTheory.Limits.Preservation.
Require Import UniMath.CategoryTheory.Limits.Examples.UnitCategoryLimits.
Require Import UniMath.CategoryTheory.Limits.Examples.CategoryProductLimits.
Require Import UniMath.CategoryTheory.Limits.Examples.IsoCommaLimits.
Require Import UniMath.CategoryTheory.Limits.Examples.EilenbergMooreLimits.
Require Import UniMath.Bicategories.Core.Bicat.
Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats.
Require Import UniMath.Bicategories.Core.Examples.StructuredCategories.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.MonadsLax.
Require Import UniMath.Bicategories.Limits.Final.
Require Import UniMath.Bicategories.Limits.Products.
Require Import UniMath.Bicategories.Limits.Pullbacks.
Require Import UniMath.Bicategories.Limits.EilenbergMooreObjects.
Require Import UniMath.Bicategories.Limits.Examples.BicatOfUnivCatsLimits.
Require Import UniMath.Bicategories.Limits.Examples.TotalBicategoryLimits.
Require Import UniMath.Bicategories.Limits.Examples.DispConstructionsLimits.
Require Import UniMath.Bicategories.Limits.Examples.SubbicatLimits.
Require Import UniMath.Bicategories.Monads.Examples.MonadsInTotalBicat.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.MonadInclusion.
Local Open Scope cat.
(**
1. Limits of categories with a terminal objects
*)
Definition disp_bifinal_univ_cat_with_terminal_obj
: disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats.disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats
Proof .disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats
use subbicat_disp_final. (λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1 bifinal_cats)
- (λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1 bifinal_cats)
exact terminal_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) terminal_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intros C.C : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr1 C) (pr1 bifinal_cats)
(pr12 C) terminal_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 C))
apply functor_to_unit_preserves_terminal.
Defined .
Definition bifinal_obj_univ_cat_with_terminal_obj
: bifinal_obj univ_cat_with_terminal_obj.bifinal_obj univ_cat_with_terminal_obj
Proof .bifinal_obj univ_cat_with_terminal_obj
use subbicat_final. bifinal_obj bicat_of_univ_cats
- bifinal_obj bicat_of_univ_cats
exact bifinal_cats.
- (λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1 bifinal_cats)
exact terminal_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) terminal_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intros C.C : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr1 C) (pr1 bifinal_cats)
(pr12 C) terminal_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 C))
apply functor_to_unit_preserves_terminal.
Defined .
Definition disp_has_binprod_univ_cat_with_terminal_obj
: disp_has_binprod disp_bicat_terminal_obj has_binprod_bicat_of_univ_cats.disp_has_binprod disp_bicat_terminal_obj
has_binprod_bicat_of_univ_cats
Proof .disp_has_binprod disp_bicat_terminal_obj
has_binprod_bicat_of_univ_cats
use subbicat_disp_binprod. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)),
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)),
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂ , terminal_category_binproduct (pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_terminal.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_terminal.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_terminal_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
preserves_terminal
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
preserves_terminal
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
preserves_terminal
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition has_binprod_univ_cat_with_terminal_obj
: has_binprod univ_cat_with_terminal_obj.has_binprod univ_cat_with_terminal_obj
Proof .has_binprod univ_cat_with_terminal_obj
use subbicat_binprod. has_binprod bicat_of_univ_cats
- has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)),
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂ , terminal_category_binproduct (pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_terminal.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_terminal.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF HG)),
terminal_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_terminal_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
preserves_terminal
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
preserves_terminal
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG) q : binprod_cone C₁ C₂
preserves_terminal
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition disp_has_pb_univ_cat_with_terminal_obj
: disp_has_pb disp_bicat_terminal_obj has_pb_bicat_of_univ_cats.disp_has_pb disp_bicat_terminal_obj
has_pb_bicat_of_univ_cats
Proof .disp_has_pb disp_bicat_terminal_obj
has_pb_bicat_of_univ_cats
use subbicat_disp_has_pb. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
exact (λ C₁ C₂ C₃ F G ,
terminal_category_iso_comma
_ _
(pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₂, C₃ ⟧),
terminal_category_iso_comma (pr1 (pr1 F,, pr12 F))
(pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G ,
iso_comma_pr1_preserves_terminal
_ _
(pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₂, C₃ ⟧),
terminal_category_iso_comma (pr1 (pr1 F,, pr12 F))
(pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G ,
iso_comma_pr2_preserves_terminal
_ _
(pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧)
(q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₂, C₃ ⟧),
terminal_category_iso_comma (pr1 (pr1 F,, pr12 F))
(pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz
g0),
composition_preserves_terminal HF HG)) q))
exact (λ C₁ C₂ C₃ F G q ,
iso_comma_ump1_preserves_terminal
_ _
(pr22 G)
_ (pr22 (pb_cone_pr1 q))
_ (pr22 (pb_cone_pr2 q))
_).
Defined .
Definition has_pb_univ_cat_with_terminal_obj
: has_pb univ_cat_with_terminal_obj.has_pb univ_cat_with_terminal_obj
Proof .has_pb univ_cat_with_terminal_obj
use subbicat_has_pb. has_pb bicat_of_univ_cats
- has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
exact (λ C₁ C₂ C₃ F G ,
terminal_category_iso_comma
_ _
(pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₂, C₃ ⟧),
terminal_category_iso_comma (pr1 (pr1 F,, pr12 F))
(pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G ,
iso_comma_pr1_preserves_terminal
_ _
(pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₂, C₃ ⟧),
terminal_category_iso_comma (pr1 (pr1 F,, pr12 F))
(pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G ,
iso_comma_pr2_preserves_terminal
_ _
(pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z Py Pz g),
composition_preserves_terminal HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz g),
composition_preserves_terminal HF HG) ⟦ y, z ⟧)
(q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F)
y0 z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_terminal F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_terminal F0) y0
z0 Py Pz g0),
composition_preserves_terminal HF HG)
⟦ C₂, C₃ ⟧),
terminal_category_iso_comma (pr1 (pr1 F,, pr12 F))
(pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G)
(pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y0 z0 Py Pz
g0),
composition_preserves_terminal HF HG)) q))
exact (λ C₁ C₂ C₃ F G q ,
iso_comma_ump1_preserves_terminal
_ _
(pr22 G)
_ (pr22 (pb_cone_pr1 q))
_ (pr22 (pb_cone_pr2 q))
_).
Defined .
Definition has_em_univ_cat_with_terminal_obj
: bicat_has_em univ_cat_with_terminal_obj.bicat_has_em univ_cat_with_terminal_obj
Proof .bicat_has_em univ_cat_with_terminal_obj
use subbicat_has_em. bicat_has_em bicat_of_univ_cats
- bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))),
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
exact (λ m ,
terminal_eilenberg_moore_cat _ (pr12 (ob_of_mnd m))).
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr11 m)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF
HG))),
terminal_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0))) m) (pr121 m)
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))))
exact (λ m ,
eilenberg_moore_pr_preserves_terminal _ (pr12 (ob_of_mnd m))).
- ∏
(m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))))
(q : em_cone m),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_terminal F) x y
Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_terminal F) y z
Py Pz g),
composition_preserves_terminal HF
HG))),
terminal_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)) m q))
intros m q.m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))) q : em_cone m
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px
Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py
Pz g),
composition_preserves_terminal HF
HG))),
terminal_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m))
(pr12 (ob_of_mnd m))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG)) m q))
use functor_to_eilenberg_moore_cat_preserves_terminal. m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F) y z Py Pz g),
composition_preserves_terminal HF HG))) q : em_cone m
preserves_terminal
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Terminal (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_terminal F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Terminal (pr1 C0)) C),
identity_preserves_terminal C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Terminal (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_terminal F) y
z Py Pz g),
composition_preserves_terminal HF HG))
m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))).
Defined .
(**
2. Limits of categories with products
*)
Definition disp_bifinal_obj_univ_cat_with_binprod
: disp_bifinal_obj disp_bicat_binprod bifinal_cats.disp_bifinal_obj disp_bicat_binprod bifinal_cats
Proof .disp_bifinal_obj disp_bicat_binprod bifinal_cats
use subbicat_disp_final. (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1 bifinal_cats)
- (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1 bifinal_cats)
exact binproduct_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) binproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro .x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) binproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
apply functor_to_unit_preserves_binproduct.
Defined .
Definition bifinal_obj_univ_cat_with_binprod
: bifinal_obj univ_cat_with_binprod.bifinal_obj univ_cat_with_binprod
Proof .bifinal_obj univ_cat_with_binprod
use subbicat_final. bifinal_obj bicat_of_univ_cats
- bifinal_obj bicat_of_univ_cats
exact bifinal_cats.
- (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1 bifinal_cats)
exact binproduct_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) binproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro .x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) binproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
apply functor_to_unit_preserves_binproduct.
Defined .
Definition disp_has_binprod_univ_cat_with_binprod
: disp_has_binprod disp_bicat_binprod has_binprod_bicat_of_univ_cats.disp_has_binprod disp_bicat_binprod
has_binprod_bicat_of_univ_cats
Proof .disp_has_binprod disp_bicat_binprod
has_binprod_bicat_of_univ_cats
use subbicat_disp_binprod. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)),
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)),
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
apply binproducts_in_product_category.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_binproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_binproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_binproduct_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
preserves_binproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
preserves_binproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
preserves_binproduct
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition has_binprod_univ_cat_with_binprod
: has_binprod univ_cat_with_binprod.has_binprod univ_cat_with_binprod
Proof .has_binprod univ_cat_with_binprod
use subbicat_binprod. has_binprod bicat_of_univ_cats
- has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)),
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
apply binproducts_in_product_category.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_binproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_binproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)),
binproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_binproduct_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
preserves_binproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
preserves_binproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) q : binprod_cone C₁ C₂
preserves_binproduct
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition disp_has_pb_univ_cat_with_binprod
: disp_has_pb disp_bicat_binprod has_pb_bicat_of_univ_cats.disp_has_pb disp_bicat_binprod
has_pb_bicat_of_univ_cats
Proof .disp_has_pb disp_bicat_binprod
has_pb_bicat_of_univ_cats
use subbicat_disp_has_pb. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧),
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧),
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply binproducts_in_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y0
z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_binproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y0
z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_binproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧) (q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y0
z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz
g0),
composition_preserves_binproduct HF HG)) q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)) q))
apply iso_comma_ump1_preserves_binproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct (pr1 G)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinProducts (pr1 C₁))
(_ : BinProducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinProducts (pr1 C)),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinProducts (pr1 x))
(_ : BinProducts (pr1 y))
(_ : BinProducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_binproduct f)
(HG : preserves_binproduct g),
composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinProducts (pr1 C₁))
(_ : BinProducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinProducts (pr1 C)),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinProducts (pr1 x))
(_ : BinProducts (pr1 y))
(_ : BinProducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_binproduct f)
(HG : preserves_binproduct g),
composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_pb_univ_cat_with_binprod
: has_pb univ_cat_with_binprod.has_pb univ_cat_with_binprod
Proof .has_pb univ_cat_with_binprod
use subbicat_has_pb. has_pb bicat_of_univ_cats
- has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧),
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply binproducts_in_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y0
z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_binproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y0
z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_binproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧
BinProducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z Py Pz g),
composition_preserves_binproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ x, z
⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz g),
composition_preserves_binproduct HF HG) ⟦ y, z
⟧) (q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y0
z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y0 z0 Py Pz g0),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y0 z0 Py Pz
g0),
composition_preserves_binproduct HF HG)) q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F) y
z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_binproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_binproduct F0)
y z Py Pz g),
composition_preserves_binproduct HF HG)
⟦ C₂, C₃ ⟧),
binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)) q))
apply iso_comma_ump1_preserves_binproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct (pr1 G)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinProducts (pr1 C₁))
(_ : BinProducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinProducts (pr1 C)),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinProducts (pr1 x))
(_ : BinProducts (pr1 y))
(_ : BinProducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_binproduct f)
(HG : preserves_binproduct g),
composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_binproduct
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinProducts (pr1 C₁))
(_ : BinProducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinProducts (pr1 C)),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinProducts (pr1 x))
(_ : BinProducts (pr1 y))
(_ : BinProducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_binproduct f)
(HG : preserves_binproduct g),
composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_em_univ_cat_with_binprod
: bicat_has_em univ_cat_with_binprod.bicat_has_em univ_cat_with_binprod
Proof .bicat_has_em univ_cat_with_binprod
use subbicat_has_em. bicat_has_em bicat_of_univ_cats
- bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz
g),
composition_preserves_binproduct HF HG))),
(λ C : bicat_of_univ_cats, BinProducts (pr1 C))
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
exact (λ m ,
BinProducts_eilenberg_moore_cat _ (pr12 (ob_of_mnd m))).
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz
g),
composition_preserves_binproduct HF HG))),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr11 m)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y
Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z
Py Pz g),
composition_preserves_binproduct
HF HG))),
BinProducts_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0))) m) (pr121 m)
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))))
exact (λ m ,
eilenberg_moore_pr_preserves_binproduct _ (pr12 (ob_of_mnd m))).
- ∏
(m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))))
(q : em_cone m),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y
Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z
Py Pz g),
composition_preserves_binproduct
HF HG))),
BinProducts_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)) m
q))
intros m q.m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz
g),
composition_preserves_binproduct HF HG))) q : em_cone m
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y
Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z
Py Pz g),
composition_preserves_binproduct HF
HG))),
BinProducts_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m))
(pr12 (ob_of_mnd m))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz g),
composition_preserves_binproduct HF HG)) m
q))
use functor_to_eilenberg_moore_cat_preserves_binproduct. m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F) y z Py Pz
g),
composition_preserves_binproduct HF HG))) q : em_cone m
preserves_binproduct
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinProducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_binproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinProducts (pr1 C0)) C),
identity_preserves_binproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinProducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_binproduct F)
y z Py Pz g),
composition_preserves_binproduct HF HG))
m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))).
Defined .
(**
3. Limits of categories with pullbacks
*)
Definition disp_bifinal_obj_univ_cat_with_pb
: disp_bifinal_obj disp_bicat_pullback bifinal_cats.disp_bifinal_obj disp_bicat_pullback bifinal_cats
Proof .disp_bifinal_obj disp_bicat_pullback bifinal_cats
use subbicat_disp_final. (λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1 bifinal_cats)
- (λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1 bifinal_cats)
exact pullbacks_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) pullbacks_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro .x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) pullbacks_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
apply functor_to_unit_preserves_pullback.
Defined .
Definition bifinal_obj_univ_cat_with_pb
: bifinal_obj univ_cat_with_pb.bifinal_obj univ_cat_with_pb
Proof .bifinal_obj univ_cat_with_pb
use subbicat_final. bifinal_obj bicat_of_univ_cats
- bifinal_obj bicat_of_univ_cats
exact bifinal_cats.
- (λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1 bifinal_cats)
exact pullbacks_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) pullbacks_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro .x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) pullbacks_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
apply functor_to_unit_preserves_pullback.
Defined .
Definition disp_has_binprod_univ_cat_with_pb
: disp_has_binprod disp_bicat_pullback has_binprod_bicat_of_univ_cats.disp_has_binprod disp_bicat_pullback
has_binprod_bicat_of_univ_cats
Proof .disp_has_binprod disp_bicat_pullback
has_binprod_bicat_of_univ_cats
use subbicat_disp_binprod. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)),
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)),
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
apply pullbacks_in_product_category.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_pullback.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_pullback.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_pullback_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
preserves_pullback
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
preserves_pullback
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
preserves_pullback
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition has_binprod_univ_cat_with_pb
: has_binprod univ_cat_with_pb.has_binprod univ_cat_with_pb
Proof .has_binprod univ_cat_with_pb
use subbicat_binprod. has_binprod bicat_of_univ_cats
- has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)),
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
apply pullbacks_in_product_category.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_pullback.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_pullback.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)),
pullbacks_in_product_category (pr12 C₁) (pr12 C₂))
C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_pullback_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
preserves_pullback
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
preserves_pullback
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) q : binprod_cone C₁ C₂
preserves_pullback
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition disp_has_pb_univ_cat_with_pb
: disp_has_pb disp_bicat_pullback has_pb_bicat_of_univ_cats.disp_has_pb disp_bicat_pullback
has_pb_bicat_of_univ_cats
Proof .disp_has_pb disp_bicat_pullback
has_pb_bicat_of_univ_cats
use subbicat_disp_has_pb. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply pullbacks_in_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y
z Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_pullback.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y
z Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_pullback.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧)
(q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz
g0),
composition_preserves_pullback HF HG)) q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y
z Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)) q))
apply iso_comma_ump1_preserves_pullback.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback (pr1 G)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C₁))
(_ : Pullbacks (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C)),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(_ : Pullbacks (pr1 x))
(_ : Pullbacks (pr1 y))
(_ : Pullbacks (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_pullback f)
(HG : preserves_pullback g),
composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C₁))
(_ : Pullbacks (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C)),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(_ : Pullbacks (pr1 x))
(_ : Pullbacks (pr1 y))
(_ : Pullbacks (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_pullback f)
(HG : preserves_pullback g),
composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_pb_univ_cat_with_pb
: has_pb univ_cat_with_pb.
Proof .
use subbicat_has_pb. has_pb bicat_of_univ_cats
- has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply pullbacks_in_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y
z Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_pullback.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y
z Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_pullback.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧
Pullbacks (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z Py Pz g),
composition_preserves_pullback HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz g),
composition_preserves_pullback HF HG) ⟦ y, z ⟧)
(q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F)
y0 z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y0
z0 Py Pz g0),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y0 z0 Py Pz
g0),
composition_preserves_pullback HF HG)) q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_pullback F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) x
y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_pullback F0) y
z Py Pz g),
composition_preserves_pullback HF HG)
⟦ C₂, C₃ ⟧),
pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F)
(pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)) q))
apply iso_comma_ump1_preserves_pullback.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback (pr1 G)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C₁))
(_ : Pullbacks (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C)),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(_ : Pullbacks (pr1 x))
(_ : Pullbacks (pr1 y))
(_ : Pullbacks (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_pullback f)
(HG : preserves_pullback g),
composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₁, C₃
⟧ G : subbicat
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG) ⟦ C₂, C₃
⟧ q : pb_cone F G
preserves_pullback
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C₁))
(_ : Pullbacks (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : Pullbacks (pr1 C)),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(_ : Pullbacks (pr1 x))
(_ : Pullbacks (pr1 y))
(_ : Pullbacks (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_pullback f)
(HG : preserves_pullback g),
composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_em_univ_cat_with_pb
: bicat_has_em univ_cat_with_pb.bicat_has_em univ_cat_with_pb
Proof .bicat_has_em univ_cat_with_pb
use subbicat_has_em. bicat_has_em bicat_of_univ_cats
- bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))),
(λ C : bicat_of_univ_cats, Pullbacks (pr1 C))
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
exact (λ m ,
Pullbacks_eilenberg_moore _ (pr12 (ob_of_mnd m))).
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr11 m)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF
HG))),
Pullbacks_eilenberg_moore
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0))) m) (pr121 m)
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))))
exact (λ m ,
eilenberg_moore_pr_preserves_pullback _ (pr12 (ob_of_mnd m))).
- ∏
(m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))))
(q : em_cone m),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_pullback F) x y
Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_pullback F) y z
Py Pz g),
composition_preserves_pullback HF
HG))),
Pullbacks_eilenberg_moore
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)) m q))
intros m q.m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))) q : em_cone m
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px
Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py
Pz g),
composition_preserves_pullback HF
HG))),
Pullbacks_eilenberg_moore
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m))
(pr12 (ob_of_mnd m))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG)) m q))
use functor_to_eilenberg_moore_cat_preserves_pullback. m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F) y z Py Pz g),
composition_preserves_pullback HF HG))) q : em_cone m
preserves_pullback
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Pullbacks (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_pullback F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Pullbacks (pr1 C0)) C),
identity_preserves_pullback C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Pullbacks (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_pullback F) y
z Py Pz g),
composition_preserves_pullback HF HG))
m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))).
Defined .
(**
4. Limits of categories with finite limits
*)
Definition disp_bifinal_obj_univ_cat_with_finlim
: disp_bifinal_obj disp_bicat_finlim bifinal_cats.disp_bifinal_obj disp_bicat_finlim bifinal_cats
Proof .disp_bifinal_obj disp_bicat_finlim bifinal_cats
use disp_dirprod_bifinal. disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats
- disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats
exact disp_bifinal_univ_cat_with_terminal_obj.
- disp_bifinal_obj disp_bicat_pullback bifinal_cats
exact disp_bifinal_obj_univ_cat_with_pb.
Defined .
Definition bifinal_obj_bicat_of_univ_cat_with_finlim
: bifinal_obj bicat_of_univ_cat_with_finlim.bifinal_obj bicat_of_univ_cat_with_finlim
Proof .bifinal_obj bicat_of_univ_cat_with_finlim
use total_bicat_final. disp_2cells_isaprop disp_bicat_finlim
- disp_2cells_isaprop disp_bicat_finlim
use disp_2cells_isaprop_prod. disp_2cells_isaprop disp_bicat_terminal_obj
+ disp_2cells_isaprop disp_bicat_terminal_obj
apply disp_2cells_isaprop_subbicat.
+ disp_2cells_isaprop disp_bicat_pullback
apply disp_2cells_isaprop_subbicat.
- ∏ (x y : bicat_of_univ_cats)
(f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g)
(xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y)
(ff : xx -->[ f] yy) (gg : xx -->[ g] yy),
disp_2cells α ff gg
intros .x, y : bicat_of_univ_cats f, g : bicat_of_univ_cats ⟦ x, y ⟧ α : f ==> g xx : disp_bicat_finlim x yy : disp_bicat_finlim y ff : xx -->[ f] yy gg : xx -->[ g] yy
disp_2cells α ff gg
exact ((tt ,, tt) ,, (tt ,, tt)).
- bifinal_obj bicat_of_univ_cats
exact bifinal_cats.
- disp_bifinal_obj disp_bicat_finlim bifinal_cats
exact disp_bifinal_obj_univ_cat_with_finlim.
Defined .
Definition disp_has_binprod_univ_cat_with_finlim
: disp_has_binprod disp_bicat_finlim has_binprod_bicat_of_univ_cats.disp_has_binprod disp_bicat_finlim
has_binprod_bicat_of_univ_cats
Proof .disp_has_binprod disp_bicat_finlim
has_binprod_bicat_of_univ_cats
use disp_dirprod_binprod. disp_has_binprod disp_bicat_terminal_obj
has_binprod_bicat_of_univ_cats
- disp_has_binprod disp_bicat_terminal_obj
has_binprod_bicat_of_univ_cats
exact disp_has_binprod_univ_cat_with_terminal_obj.
- disp_has_binprod disp_bicat_pullback
has_binprod_bicat_of_univ_cats
exact disp_has_binprod_univ_cat_with_pb.
Defined .
Definition has_binprod_bicat_of_univ_cat_with_finlim
: has_binprod bicat_of_univ_cat_with_finlim.has_binprod bicat_of_univ_cat_with_finlim
Proof .has_binprod bicat_of_univ_cat_with_finlim
use total_bicat_prod. disp_2cells_isaprop disp_bicat_finlim
- disp_2cells_isaprop disp_bicat_finlim
use disp_2cells_isaprop_prod. disp_2cells_isaprop disp_bicat_terminal_obj
+ disp_2cells_isaprop disp_bicat_terminal_obj
apply disp_2cells_isaprop_subbicat.
+ disp_2cells_isaprop disp_bicat_pullback
apply disp_2cells_isaprop_subbicat.
- ∏ (x y : bicat_of_univ_cats)
(f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g)
(xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y)
(ff : xx -->[ f] yy) (gg : xx -->[ g] yy),
disp_2cells α ff gg
intros .x, y : bicat_of_univ_cats f, g : bicat_of_univ_cats ⟦ x, y ⟧ α : f ==> g xx : disp_bicat_finlim x yy : disp_bicat_finlim y ff : xx -->[ f] yy gg : xx -->[ g] yy
disp_2cells α ff gg
exact ((tt ,, tt) ,, (tt ,, tt)).
- disp_locally_groupoid disp_bicat_finlim
apply disp_locally_groupoid_prod.disp_locally_groupoid disp_bicat_terminal_obj
+ disp_locally_groupoid disp_bicat_terminal_obj
apply disp_locally_groupoid_subbicat.is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.
+ disp_locally_groupoid disp_bicat_pullback
apply disp_locally_groupoid_subbicat.is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.
- has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.
- disp_has_binprod disp_bicat_finlim
has_binprod_bicat_of_univ_cats
exact disp_has_binprod_univ_cat_with_finlim.
Defined .
Definition disp_has_pb_univ_cat_with_finlim
: disp_has_pb disp_bicat_finlim has_pb_bicat_of_univ_cats.disp_has_pb disp_bicat_finlim
has_pb_bicat_of_univ_cats
Proof .disp_has_pb disp_bicat_finlim
has_pb_bicat_of_univ_cats
use disp_dirprod_pb. disp_has_pb disp_bicat_terminal_obj
has_pb_bicat_of_univ_cats
- disp_has_pb disp_bicat_terminal_obj
has_pb_bicat_of_univ_cats
exact disp_has_pb_univ_cat_with_terminal_obj.
- disp_has_pb disp_bicat_pullback
has_pb_bicat_of_univ_cats
exact disp_has_pb_univ_cat_with_pb.
Defined .
Definition has_pb_bicat_of_univ_cat_with_finlim
: has_pb bicat_of_univ_cat_with_finlim.has_pb bicat_of_univ_cat_with_finlim
Proof .has_pb bicat_of_univ_cat_with_finlim
use total_bicat_has_pb. disp_2cells_isaprop disp_bicat_finlim
- disp_2cells_isaprop disp_bicat_finlim
use disp_2cells_isaprop_prod. disp_2cells_isaprop disp_bicat_terminal_obj
+ disp_2cells_isaprop disp_bicat_terminal_obj
apply disp_2cells_isaprop_subbicat.
+ disp_2cells_isaprop disp_bicat_pullback
apply disp_2cells_isaprop_subbicat.
- ∏ (x y : bicat_of_univ_cats)
(f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g)
(xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y)
(ff : xx -->[ f] yy) (gg : xx -->[ g] yy),
disp_2cells α ff gg
intros .x, y : bicat_of_univ_cats f, g : bicat_of_univ_cats ⟦ x, y ⟧ α : f ==> g xx : disp_bicat_finlim x yy : disp_bicat_finlim y ff : xx -->[ f] yy gg : xx -->[ g] yy
disp_2cells α ff gg
exact ((tt ,, tt) ,, (tt ,, tt)).
- disp_locally_groupoid disp_bicat_finlim
apply disp_locally_groupoid_prod.disp_locally_groupoid disp_bicat_terminal_obj
+ disp_locally_groupoid disp_bicat_terminal_obj
apply disp_locally_groupoid_subbicat.is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.
+ disp_locally_groupoid disp_bicat_pullback
apply disp_locally_groupoid_subbicat.is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.
- has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.
- disp_has_pb disp_bicat_finlim
has_pb_bicat_of_univ_cats
exact disp_has_pb_univ_cat_with_finlim.
Defined .
(**
5. Limits of categories with initial objects
*)
Definition disp_bifinal_obj_univ_cat_with_initial
: disp_bifinal_obj disp_bicat_initial_obj bifinal_cats.disp_bifinal_obj disp_bicat_initial_obj bifinal_cats
Proof .disp_bifinal_obj disp_bicat_initial_obj bifinal_cats
use subbicat_disp_final. (λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1 bifinal_cats)
- (λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1 bifinal_cats)
exact initial_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) initial_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro C.C : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr1 C) (pr1 bifinal_cats)
(pr12 C) initial_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 C))
apply functor_to_unit_preserves_initial.
Defined .
Definition bifinal_obj_univ_cat_with_initial
: bifinal_obj univ_cat_with_initial.bifinal_obj univ_cat_with_initial
Proof .bifinal_obj univ_cat_with_initial
use subbicat_final. bifinal_obj bicat_of_univ_cats
- bifinal_obj bicat_of_univ_cats
exact bifinal_cats.
- (λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1 bifinal_cats)
exact initial_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) initial_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro C.C : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr1 C) (pr1 bifinal_cats)
(pr12 C) initial_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 C))
apply functor_to_unit_preserves_initial.
Defined .
Definition disp_has_binprod_univ_cat_with_initial
: disp_has_binprod disp_bicat_initial_obj has_binprod_bicat_of_univ_cats.disp_has_binprod disp_bicat_initial_obj
has_binprod_bicat_of_univ_cats
Proof .disp_has_binprod disp_bicat_initial_obj
has_binprod_bicat_of_univ_cats
use subbicat_disp_binprod. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)),
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)),
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂ , initial_category_binproduct (pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0 z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂)) x
y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_initial.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0 z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂)) x
y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_initial.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0 z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂)) x
y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_initial_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
preserves_initial
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
preserves_initial
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
preserves_initial
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition has_binprod_univ_cat_with_initial
: has_binprod univ_cat_with_initial.has_binprod univ_cat_with_initial
Proof .has_binprod univ_cat_with_initial
use subbicat_binprod. has_binprod bicat_of_univ_cats
- has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)),
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂ , initial_category_binproduct (pr12 C₁) (pr12 C₂)).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0 z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂)) x
y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_initial.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0 z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂)) x
y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_initial.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0 z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂)) x
y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG)),
initial_category_binproduct (pr12 C₁) (pr12 C₂))
C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_initial_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
preserves_initial
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
preserves_initial
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) q : binprod_cone C₁ C₂
preserves_initial
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition disp_has_pb_univ_cat_with_initial
: disp_has_pb disp_bicat_initial_obj has_pb_bicat_of_univ_cats.disp_has_pb disp_bicat_initial_obj
has_pb_bicat_of_univ_cats
Proof .disp_has_pb disp_bicat_initial_obj
has_pb_bicat_of_univ_cats
use subbicat_disp_has_pb. ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply initial_category_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0 y0
Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0 z0
Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y0
z0 Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_initial.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0 y0
Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0 z0
Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y0
z0 Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_initial.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧)
(q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0 y0
Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0 z0
Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y0
z0 Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g0),
composition_preserves_initial HF HG)) q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)) q))
apply iso_comma_ump1_preserves_initial.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Initial (pr1 C₁))
(_ : Initial (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : Initial (pr1 C)),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(_ : Initial (pr1 x))
(_ : Initial (pr1 y))
(_ : Initial (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_initial f)
(HG : preserves_initial g),
composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Initial (pr1 C₁))
(_ : Initial (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : Initial (pr1 C)),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(_ : Initial (pr1 x))
(_ : Initial (pr1 y))
(_ : Initial (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_initial f)
(HG : preserves_initial g),
composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_pb_univ_cat_with_initial
: has_pb univ_cat_with_initial.has_pb univ_cat_with_initial
Proof .has_pb univ_cat_with_initial
use subbicat_has_pb. has_pb bicat_of_univ_cats
- has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧),
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply initial_category_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0 y0
Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0 z0
Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y0
z0 Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_initial.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0 y0
Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0 z0
Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y0
z0 Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_initial.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z Py Pz g),
composition_preserves_initial HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ x, z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g),
composition_preserves_initial HF HG) ⟦ y, z ⟧)
(q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0
z0 Py Pz g0),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x0 y0
Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y0 z0
Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x0
y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y0
z0 Py Pz g0),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x0 y0 Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y0 z0 Py Pz g0),
composition_preserves_initial HF HG)) q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁,
C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_initial F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) x y
Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_initial F0) y z
Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂,
C₃ ⟧),
initial_category_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)) q))
apply iso_comma_ump1_preserves_initial.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Initial (pr1 C₁))
(_ : Initial (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : Initial (pr1 C)),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(_ : Initial (pr1 x))
(_ : Initial (pr1 y))
(_ : Initial (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_initial f)
(HG : preserves_initial g),
composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) F : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧ q : pb_cone F G
preserves_initial
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : Initial (pr1 C₁))
(_ : Initial (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : Initial (pr1 C)),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(_ : Initial (pr1 x))
(_ : Initial (pr1 y))
(_ : Initial (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_initial f)
(HG : preserves_initial g),
composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_em_univ_cat_with_initial
: bicat_has_em univ_cat_with_initial.bicat_has_em univ_cat_with_initial
Proof .bicat_has_em univ_cat_with_initial
use subbicat_has_em. bicat_has_em bicat_of_univ_cats
- bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))),
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
refine (λ m ,
initial_eilenberg_moore_cat
_
(pr12 (ob_of_mnd m))
_).m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)))
preserves_initial
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr11 m)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px
Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py
Pz g),
composition_preserves_initial HF
HG))),
initial_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m)
(pr121 m)
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))))
refine (λ m ,
eilenberg_moore_pr_preserves_initial _ (pr12 (ob_of_mnd m)) _).m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)))
preserves_initial
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
- ∏
(m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x y
Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y z
Py Pz g),
composition_preserves_initial HF HG))))
(q : em_cone m),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px
Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py
Pz g),
composition_preserves_initial HF
HG))),
initial_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)) m q))
intros m q.m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))) q : em_cone m
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px
Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py
Pz g),
composition_preserves_initial HF HG))),
initial_eilenberg_moore_cat
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m))
(pr12 (ob_of_mnd m)) (pr22 (endo_of_mnd m))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG)) m q))
use functor_to_eilenberg_moore_cat_preserves_initial. m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))) q : em_cone m
preserves_initial
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
+ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))) q : em_cone m
preserves_initial
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
+ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats, Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F) y z Py Pz g),
composition_preserves_initial HF HG))) q : em_cone m
preserves_initial
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
Initial (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_initial F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
Initial (pr1 C0)) C),
identity_preserves_initial C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) x
y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
Initial (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_initial F) y
z Py Pz g),
composition_preserves_initial HF HG)) m
q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))).
Defined .
(**
6. Limits of categories with coproducts
*)
Definition disp_bifinal_obj_univ_cat_with_bincoprod
: disp_bifinal_obj disp_bicat_bincoprod bifinal_cats.disp_bifinal_obj disp_bicat_bincoprod bifinal_cats
Proof .disp_bifinal_obj disp_bicat_bincoprod bifinal_cats
use subbicat_disp_final. (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1 bifinal_cats)
- (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1 bifinal_cats)
exact bincoproduct_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) bincoproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro C.C : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr1 C) (pr1 bifinal_cats)
(pr12 C) bincoproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 C))
apply functor_to_unit_preserves_bincoproduct.
Defined .
Definition bifinal_obj_univ_cat_with_bincoprod
: bifinal_obj univ_cat_with_bincoprod.bifinal_obj univ_cat_with_bincoprod
Proof .bifinal_obj univ_cat_with_bincoprod
use subbicat_final. bifinal_obj bicat_of_univ_cats
- bifinal_obj bicat_of_univ_cats
exact bifinal_cats.
- (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1 bifinal_cats)
exact bincoproduct_unit_category.
- ∏
x : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr1 x) (pr1 bifinal_cats)
(pr12 x) bincoproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 x))
intro C.C : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr1 C) (pr1 bifinal_cats)
(pr12 C) bincoproduct_unit_category
(is_bifinal_1cell_property (pr2 bifinal_cats)
(pr1 C))
apply functor_to_unit_preserves_bincoproduct.
Defined .
Definition disp_has_binprod_univ_cat_with_bincoprod
: disp_has_binprod disp_bicat_bincoprod has_binprod_bicat_of_univ_cats.disp_has_binprod disp_bicat_bincoprod
has_binprod_bicat_of_univ_cats
Proof .disp_has_binprod disp_bicat_bincoprod
has_binprod_bicat_of_univ_cats
use subbicat_disp_binprod. ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
apply bincoproducts_in_product_category.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z
Py Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_bincoproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z
Py Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_bincoproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z
Py Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_bincoproduct_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
preserves_bincoproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
preserves_bincoproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
preserves_bincoproduct
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition has_binprod_univ_cat_with_bincoprod
: has_binprod univ_cat_with_bincoprod.has_binprod univ_cat_with_bincoprod
Proof .has_binprod univ_cat_with_bincoprod
use subbicat_binprod. has_binprod bicat_of_univ_cats
- has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
apply bincoproducts_in_product_category.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z
Py Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 x)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₁)
(binprod_cone_pr1
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr1_preserves_bincoproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z
Py Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y) (pr12 y)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))))
intros C₁ C₂.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂) (pr12 C₂)
(binprod_cone_pr2
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))))
apply pr2_preserves_bincoproduct.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(q : binprod_cone x y),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z
Py Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) x y)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 x)
(pr1 y))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
intros C₁ C₂ q.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
(pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)),
bincoproducts_in_product_category (pr12 C₁)
(pr12 C₂)) C₁ C₂)
(binprod_ump_1cell
(pr2
(has_binprod_bicat_of_univ_cats (pr1 C₁)
(pr1 C₂))) (pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q)))
apply preserves_bincoproduct_bindelta_pair_functor.C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
preserves_bincoproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
preserves_bincoproduct
(binprod_cone_pr1
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
+ C₁, C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) q : binprod_cone C₁ C₂
preserves_bincoproduct
(binprod_cone_pr2
(make_binprod_cone (pr1 q)
(pr1 (binprod_cone_pr1 q))
(pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)).
Defined .
Definition disp_has_pb_univ_cat_with_bincoprod
: disp_has_pb disp_bicat_bincoprod has_pb_bicat_of_univ_cats.disp_has_pb disp_bicat_bincoprod
has_pb_bicat_of_univ_cats
Proof .disp_has_pb disp_bicat_bincoprod
has_pb_bicat_of_univ_cats
use subbicat_disp_has_pb. ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧),
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧),
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply bincoproducts_in_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z0
Py Pz g0),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_bincoproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z0
Py Pz g0),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_bincoproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧) (q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z0
Py Pz g0),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px
Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py
Pz g0),
composition_preserves_bincoproduct HF HG))
q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz
g),
composition_preserves_bincoproduct HF HG))
q))
apply iso_comma_ump1_preserves_bincoproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C₁))
(_ : BinCoproducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C)),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 x))
(_ : BinCoproducts (pr1 y))
(_ : BinCoproducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_bincoproduct f)
(HG : preserves_bincoproduct g),
composition_preserves_bincoproduct HF HG))
q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C₁))
(_ : BinCoproducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C)),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 x))
(_ : BinCoproducts (pr1 y))
(_ : BinCoproducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_bincoproduct f)
(HG : preserves_bincoproduct g),
composition_preserves_bincoproduct HF HG))
q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_pb_univ_cat_with_bincoprod
: has_pb univ_cat_with_bincoprod.has_pb univ_cat_with_bincoprod
Proof .has_pb univ_cat_with_bincoprod
use subbicat_has_pb. has_pb bicat_of_univ_cats
- has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧),
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
apply bincoproducts_in_iso_comma.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 x)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z0
Py Pz g0),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 x)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₁)
(pb_cone_pr1
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr1_preserves_bincoproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr1 y)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z0
Py Pz g0),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pr12 y)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))))
intros C₁ C₂ C₃ F G.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G) (pr12 C₂)
(pb_cone_pr2
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))))
apply iso_comma_pr2_preserves_bincoproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
preserves_bincoproduct (pr1 G)
exact (pr22 G).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₁)
exact (pr12 C₁).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧
BinCoproducts (pr1 C₂)
exact (pr12 C₂).
- ∏
(x : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(y : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(z : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z Py Pz g),
composition_preserves_bincoproduct HF HG))
(f : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ x,
z ⟧)
(g : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px Py
f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ y,
z ⟧) (q : pb_cone f g),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x0 y0
Px Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y0 z0
Py Pz g0),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x0 y0 Px Py f0)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y0 z0 Py Pz g0),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 x) (pr1 y)
(pr1 z) (pr1 f) (pr1 g)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x0 y0 z0 : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x0)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y0)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z0)
(f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧)
(g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x0 y0 Px
Py f0)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y0 z0 Py
Pz g0),
composition_preserves_bincoproduct HF HG))
q))
intros C₁ C₂ C₃ F G q.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr1
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G))) (pr121 q)
((λ (C₁ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG))
(C₂ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁0, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))
(F : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₁, C₃ ⟧)
(G : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧),
preserves_bincoproduct F0)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
x y Px Py f)
(HG : (λ (C₁0 C₂0 : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁0)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂0)
(F0 : bicat_of_univ_cats ⟦ C₁0, C₂0
⟧), preserves_bincoproduct F0)
y z Py Pz g),
composition_preserves_bincoproduct HF HG)
⟦ C₂, C₃ ⟧),
bincoproducts_in_iso_comma (pr1 F) (pr1 G)
(pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃
F G)
(pb_ump_mor
(pr2
(has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)
(pr1 C₃) (pr1 F) (pr1 G)))
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz
g),
composition_preserves_bincoproduct HF HG))
q))
apply iso_comma_ump1_preserves_bincoproduct.C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct (pr1 F)
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct (pr1 F)
exact (pr22 F).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct
(pb_cone_pr1
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C₁))
(_ : BinCoproducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C)),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 x))
(_ : BinCoproducts (pr1 y))
(_ : BinCoproducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_bincoproduct f)
(HG : preserves_bincoproduct g),
composition_preserves_bincoproduct HF HG))
q))
exact (pr22 (pb_cone_pr1 q)).
+ C₁, C₂, C₃ : subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) F : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₁,
C₃ ⟧ G : subbicat
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz g),
composition_preserves_bincoproduct HF HG) ⟦ C₂,
C₃ ⟧ q : pb_cone F G
preserves_bincoproduct
(pb_cone_pr2
(total_pb_cone_help_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C₁))
(_ : BinCoproducts (pr1 C₂))
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 C)),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(_ : BinCoproducts (pr1 x))
(_ : BinCoproducts (pr1 y))
(_ : BinCoproducts (pr1 z))
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : preserves_bincoproduct f)
(HG : preserves_bincoproduct g),
composition_preserves_bincoproduct HF HG))
q))
exact (pr22 (pb_cone_pr2 q)).
Defined .
Definition has_em_univ_cat_with_bincoprod
: bicat_has_em univ_cat_with_bincoprod.bicat_has_em univ_cat_with_bincoprod
Proof .bicat_has_em univ_cat_with_bincoprod
use subbicat_has_em. bicat_has_em bicat_of_univ_cats
- bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.
- is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG))),
(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
refine (λ m ,
bincoproducts_eilenberg_moore
_
(pr12 (ob_of_mnd m))
_).m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG)))
preserves_bincoproduct
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
- ∏
m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG))),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr11 m)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x
y Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y
z Py Pz g),
composition_preserves_bincoproduct
HF HG))),
bincoproducts_eilenberg_moore
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m)
(pr121 m)
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))))
refine (λ m ,
eilenberg_moore_pr_preserves_bincoproduct _ (pr12 (ob_of_mnd m)) _).m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG)))
preserves_bincoproduct
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
- ∏
(m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_bincoproduct F)
x y Px Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧), preserves_bincoproduct F)
y z Py Pz g),
composition_preserves_bincoproduct HF HG))))
(q : em_cone m),
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m0 : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x
y Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y
z Py Pz g),
composition_preserves_bincoproduct
HF HG))),
bincoproducts_eilenberg_moore
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m0))
(pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz
g),
composition_preserves_bincoproduct HF HG))
m q))
intros m q.m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG))) q : em_cone m
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C))
C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) (pr11 q)
(pr11 (has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m))) (pr121 q)
((λ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x
y Px Py f)
(HG : (λ (C₁
C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
C₂)
(F : bicat_of_univ_cats
⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y
z Py Pz g),
composition_preserves_bincoproduct
HF HG))),
bincoproducts_eilenberg_moore
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m))
(pr12 (ob_of_mnd m)) (pr22 (endo_of_mnd m))) m)
(em_ump_1_mor (pr1_of_mnd_total_bicat m)
(pr2
(has_em_bicat_of_univ_cats
(pr1_of_mnd_total_bicat m)))
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px Py
f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py Pz
g),
composition_preserves_bincoproduct HF HG))
m q))
use functor_to_eilenberg_moore_cat_preserves_bincoproduct. m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG))) q : em_cone m
preserves_bincoproduct
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
+ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG))) q : em_cone m
preserves_bincoproduct
(Monads.functor_from_Monad
(MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad
(pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
+ m : mnd
(total_bicat
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF HG))) q : em_cone m
preserves_bincoproduct
(mor_of_mnd_mor
(mor_of_em_cone (pr1_of_mnd_total_bicat m)
(pr1_of_em_cone
(disp_subbicat
(λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C))
(λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧),
preserves_bincoproduct F)
(λ (C : bicat_of_univ_cats)
(_ : (λ C0 : bicat_of_univ_cats,
BinCoproducts (pr1 C0)) C),
identity_preserves_bincoproduct C)
(λ (x y z : bicat_of_univ_cats)
(Px : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) x)
(Py : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) y)
(Pz : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) z)
(f : bicat_of_univ_cats ⟦ x, y ⟧)
(g : bicat_of_univ_cats ⟦ y, z ⟧)
(HF : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) x y Px
Py f)
(HG : (λ (C₁ C₂ : bicat_of_univ_cats)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₁)
(_ : (λ C : bicat_of_univ_cats,
BinCoproducts (pr1 C)) C₂)
(F : bicat_of_univ_cats ⟦ C₁, C₂
⟧),
preserves_bincoproduct F) y z Py
Pz g),
composition_preserves_bincoproduct HF
HG)) m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))).
Defined .